Q120 / 170 · Manipal (MET) 2020 · Mathematics

The value of $\tan \left[2 \tan ^{-1}\left(\frac{1}{5}\right)-\frac{\pi}{4}\right]$ is

  • A
    $\frac{17}{7}$
  • B
    $\frac{-17}{7}$
  • C
    $\frac{7}{17}$
  • Correct answer
    D
    $\frac{-7}{17}$

Solution

$\tan \left[2 \tan ^{-1}\left(\frac{1}{5}\right)-\frac{\pi}{4}\right]=\tan \left[\tan ^{-1}\left(\frac{\frac{2}{5}}{1-\frac{1}{25}}\right)-\frac{\pi}{4}\right]$ $\begin{aligned} & {\left[\because 2 \tan ^{-1} \theta=\tan ^{-1}\left(\frac{2 \theta}{1-\theta^2}\right)\right]} \\ & =\tan \left[\tan ^{-1}\left(\frac{10}{24}\right)-\frac{\pi}{4}\right] \\ & =\frac{\tan \left\{\tan ^{-1}\left(\frac{10}{24}\right)\right\}-\tan \left(\frac{\pi}{4}\right)}{1+\tan \left\{\tan ^{-1}\left(\frac{10}{24}\right)\right\} \cdot \tan \left(\frac{\pi}{4}\right)}\end{aligned}$ $\begin{aligned} & =\frac{\frac{10}{24}-1}{1+\frac{10}{24} \cdot 1} \\ & =\frac{10-24}{24+10} \\ & =\frac{-14}{34}=\frac{-7}{17}\end{aligned}$

Hence, the correct answer is option: 'D'.

Relevant Questions